## Preprints (rote Reihe) des Fachbereich Mathematik

### Refine

#### Year of publication

- 1999 (3) (remove)

#### Keywords

- average density (2)
- Brownian motion (1)
- Palm distribution (1)
- Palm distributions (1)
- Rectifiability (1)
- average densities (1)
- density distribution (1)
- geometric measure theory (1)
- intersection local time (1)
- lacunarity distribution (1)

- 303
- Density theorems for the intersection local times of planar Brownian motion (1999)
- We show that the intersection local times \(\mu_p\) on the intersection of \(p\) independent planar Brownian paths have an average density of order three with respect to the gauge function \(r^2\pi\cdot (log(1/r)/\pi)^p\), more precisely, almost surely, \[ \lim\limits_{\varepsilon\downarrow 0} \frac{1}{log |log\ \varepsilon|} \int_\varepsilon^{1/e} \frac{\mu_p(B(x,r))}{r^2\pi\cdot (log(1/r)/\pi)^p} \frac{dr}{r\ log (1/r)} = 2^p \mbox{ at $\mu_p$-almost every $x$.} \] We also show that the lacunarity distributions of \(\mu_p\), at \(\mu_p\)-almost every point, is given as the distribution of the product of \(p\) independent gamma(2)-distributed random variables. The main tools of the proof are a Palm distribution associated with the intersection local time and an approximation theorem of Le Gall.

- 295
- Tangent measure distributions of fractal measures (1999)
- Tangent measure distributions are a natural tool to describe the local geometry of arbitrary measures of any dimension. We show that for every measure on a Euclidean space and every s, at almost every point, all s-dimensional tangent measure distributions define statistically self-similar random measures. Consequently, the local geometry of general measures is not different from the local geometry of self-similar sets. We illustrate the strength of this result by showing how it can be used to improve recently proved relations between ordinary and average densities.

- 294
- Average densities and linear rectifiability of measures (1999)
- We show that a measure in a Euclidean space is linearly rectifiable if and only if the lower 1-density is positive and finite and agrees with the lower average 1-density almost everywhere.